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Theorem 00lss 16049
Description: The empty structure has no subspaces (for use with fvco4i 5830). (Contributed by Stefan O'Rear, 31-Mar-2015.)
Assertion
Ref Expression
00lss  |-  (/)  =  (
LSubSp `  (/) )

Proof of Theorem 00lss
StepHypRef Expression
1 noel 3617 . . 3  |-  -.  a  e.  (/)
2 base0 13537 . . . . . 6  |-  (/)  =  (
Base `  (/) )
3 eqid 2442 . . . . . 6  |-  ( LSubSp `  (/) )  =  ( LSubSp `
 (/) )
42, 3lssss 16044 . . . . 5  |-  ( a  e.  ( LSubSp `  (/) )  -> 
a  C_  (/) )
5 ss0 3643 . . . . 5  |-  ( a 
C_  (/)  ->  a  =  (/) )
64, 5syl 16 . . . 4  |-  ( a  e.  ( LSubSp `  (/) )  -> 
a  =  (/) )
73lssn0 16048 . . . . 5  |-  ( a  e.  ( LSubSp `  (/) )  -> 
a  =/=  (/) )
87neneqd 2623 . . . 4  |-  ( a  e.  ( LSubSp `  (/) )  ->  -.  a  =  (/) )
96, 8pm2.65i 168 . . 3  |-  -.  a  e.  ( LSubSp `  (/) )
101, 92false 341 . 2  |-  ( a  e.  (/)  <->  a  e.  (
LSubSp `  (/) ) )
1110eqriv 2439 1  |-  (/)  =  (
LSubSp `  (/) )
Colors of variables: wff set class
Syntax hints:    = wceq 1653    e. wcel 1727    C_ wss 3306   (/)c0 3613   ` cfv 5483   LSubSpclss 16039
This theorem is referenced by:  00lsp  16088  lidlval  16296
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1668  ax-8 1689  ax-13 1729  ax-14 1731  ax-6 1746  ax-7 1751  ax-11 1763  ax-12 1953  ax-ext 2423  ax-sep 4355  ax-nul 4363  ax-pow 4406  ax-pr 4432
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2291  df-mo 2292  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-ral 2716  df-rex 2717  df-rab 2720  df-v 2964  df-sbc 3168  df-dif 3309  df-un 3311  df-in 3313  df-ss 3320  df-nul 3614  df-if 3764  df-pw 3825  df-sn 3844  df-pr 3845  df-op 3847  df-uni 4040  df-br 4238  df-opab 4292  df-mpt 4293  df-id 4527  df-xp 4913  df-rel 4914  df-cnv 4915  df-co 4916  df-dm 4917  df-iota 5447  df-fun 5485  df-fv 5491  df-ov 6113  df-slot 13504  df-base 13505  df-lss 16040
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